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Courses andProjects

College of EngineeringSchool of Mathematics and Statistics

Honours 2016

SCHOOL OF MATHEMATICS ANDSTATISTICS

HONOURS 2016

Welcome to Honours! Honours is an ad-ditional year in which you study selectedmathematical and/or statistical topics indepth. There is a change of emphasisfrom the preceding undergraduate years:courses tend to be more focused on a spe-cific problem or class of problems, ratherthan attempting to give a broad coverageof a branch of Mathematics or Statisticsas was done in your earlier undergraduateyears.

Also in your Honours year, there is the pos-sibility of a project in which you investi-gate some problemwith the assistance ofa member of sta�. Depending on the na-ture of the problem, this may involve lit-erature searches in the library, the use ofvarious computing packages (for exampleMATLAB, MAPLE or R) on the school’s com-puting system, resources on the internet,or even proving new theorems. At the endof theproject, youproduceawritten reportand give an oral presentation.

Who should think about Honours? If youview Mathematics and Statistics as morethan a means to an end, then the Honoursyear will be a year well spent. The Hon-ours year provides that additional mentaledge, as well as some specific technicaland (in the project) presentational skillsthat potential employers find valuable. Ifyou intend toproceedon to graduatework,Masters or PhD, then Honours is a goodstepping-stone.

To enter Honours in Mathematics, you willneed at least 60 points from MATH310-399, plus at least 30 points from 300-levelMATH,STATorother approvedcourses. ForHonours in Statistics, you need at least 60points of 300-level STAT courses and an-other 30 points from 300-level STAT, MATHor other approved courses. Normally youshould have maintained at least a B+ av-erage in these papers. Precise details aregiven in the University Calendar. Thereis also a joint Mathematics and StatisticsHonours degree (see UC enrolment hand-book).

The School o�ers both B.A.(Hons),B.Sc.(Hons) and a P.G.Dip.Sc. programs.The most appropriate program is best de-cided on a case-by-case basis. If you de-cide that you are interested in Honours,you should see the Head of School, Prof.Jennifer Brown, or the Honours Coordi-nator Dr Mark Hickman to discuss youroptions. You should do this before theenrolment week next year.

Courses are packaged in modules that willrun at two lectures a week for a semester.Anyproposedprogrammeof study forHon-ours requires the approval of the Head ofSchool. It is highly unlikely that any pro-posed programme that has a high work-load in one semester will be approved, soyou should try to construct a programmethat balances your workload evenly overboth semesters. Courses at the appropri-

ate level from other departments may beincluded in an approved program. In addi-tion there are a number of joint programsbetween the School of Mathematics andStatistics andotherdepartmentsat theuni-versity.

B.SC.(HONS)

In the Science faculty, Honours maybe completed in Mathematics, Statistics,Mathematics and Statistics, MathematicalPhysics, Computational and Applied Math-ematics (CAMS), Mathematics and Philoso-phy or Economics and Mathematics. Stu-dents enrolled for a B.Sc.(Hons) in Math-ematics and/or Statistics are required tocomplete eight 400-level courses, as wellas a project which is worth the equivalentof two courses. If you are interested in the:

• Mathematical Physics programme,see Prof. David Wiltshire (Physics).

• CAMS programme, see Assoc. Prof.Rick Beatson.

• Mathematics and Philosophy pro-gramme, see Dr. Maarten McKubre-Jordens or Dr Clemency Montelle.

• Economics and Mathematics pro-gramme, see Dr. Rua Murray.

B.A.(HONS)

In the Arts faculty, Honours may be com-pleted in Mathematics or in Statistics. AB..A. (Hons) has the same requirements asa B.Sc.(Hons).

P.G.DIP.SCI.

A Post Graduate Diploma in Science(P.G.Dip.Sci.) can be obtained by com-pleting eight 400-level papers in one year.These courses normally do not includethe honours project (MATH/STAT449). En-try requirements are as for a B.Sc.(Hons)except that you are not required to havea B+ average. It is very strongly recom-mended that your average grade in yourmajoring subject at stage 3 is at least a C+.A P.G.Dip.Sci may be used as the first partof an M.Sc.

PROPOSED COURSES FOR 2016

Theproposedcourses for 2016areoutlinedin this booklet. The final decision aboutwhich courses are to be o�ered will de-pend on the availability of sta� and onstudent interest. The School reserves theright to cancel any course that does notattract four or more students. It is alsopossible (and in some cases desirable)to include in your Honours programmeappropriate courses from other depart-ments or 300-level courses from Mathe-matics or Statistics. In the latter case, extrawork is usually required to bring the paperto a level consistent with other 400-levelcourses. Note that, subject to having therequired background, any STAT coursesmay be included in a Mathematics degreeand vice versa.

PROPOSED PROJECTS FOR 2016

A broad range of possible projects is out-lined in this booklet. However, this list isnot exhaustive and other possibilities for

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projects are certainly possible. Project su-pervision is bymutual agreement of the su-pervisor and student. You should arrangeyour project by the end of the first weekof term in 2016. It is suggested that youseek out possible supervisors during theenrolment week (or before if you have aclear idea of what sort of project interestsyou). You will hand in a written report onSeptember 12; part of your project assess-ment will be an oral presentation in Term4.

400-LEVEL COURSES

All 400-level courses are o�ered subject tosu�icient student demand and sta� avail-ability, which will be determined at thebeginning of each semester. For an up-to-date view of the School’s current o�erings,please see the School’s web page, which islocated o� the main University web page.

MATHEMATICS

MATH401 15 pointsDynamical Systems 1MATH401-16S1 (C)Dynamical systems is a rapidly developingbranch of Mathematics with growing ap-plications in diverse fields from traditionalareas of applied mathematics to numeri-cal analysis, biological systems, economicmodels andmedicine.

It is o�en di�icult or impossible to writedown an exact solution to systems of non-linear equations. The emphasis in thiscourse will be on qualitative techniquesfor classifying the behaviour of nonlin-ear systems, without necessarily solving

them exactly. Bothmain types of dynam-ical system will be studied: discrete sys-tems, consisting of an iterated map; andcontinuous systems, consisting of nonlin-ear di�erential equations. Topics coveredwill include: chaotic behaviour of simple1D maps; period-doubling bifurcations;phase portrait analysis; methods for deter-mining stability of fixed points and limitcycles; centre manifolds; and symbolicdynamics.

Enquires: Rua Murray

MATH406 15 pointsMathematical Models in BiologyMATH406-16S1 (C)How did the leopard get its spots? Whyshould children be vaccinated againstmeasles? This course will try to answerthese (and other) questions by usingmath-ematical models to examine biologicalphenomena. In the achieving this we willstudy: Biochemical reactions, reaction dif-fusion, cellular homeostasis, membraneion channels, excitability and nonlinearwave propagation.

Some knowledge of dynamical systemsand a familiarity with MAPLE (or MATLAB)are useful pre-requisites to this course.

Enquires: Alex James or Phil Wilson

MATH407 15 pointsSpecial Topic in Mathematics(Stochastic Processes)MATH407-16S2 (C)Term3: (MikeSteel) Probability theoryRan-dom graphs; probabilistic method Branch-ing processes, Polya-urn models, martin-gales

Term 4: (Michael Plank) Discrete random

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walks Continuous random walks Pois-son and pure-birth processes Markov pro-cesses

Enquires: Michael Plank or Mike Steel

MATH409 15 pointsCryptography and Coding Theory IIMATH409-16S1 (C)The purpose of this course is to study appli-cations ofNumber Theory toCryptographyand Coding Theory. These topics addressthe problem of preserving data integrityduring transmission or storage against ma-licious attacks, in the first case and againstcorruption due to random noise in the sec-ond. The main ideas will be introducedand then methods to perform e�icientlythe number theoretic calculations thatcome up in cryptography and coding the-ory will be studied. In particular, we willcover the use of elliptic curves in cryptog-raphy.

Enquires: Felipe Voloch

MATH410 15 pointsApproximation TheoryMATH410-16S1 (C)Approximation theory lies at the interfaceof many specialties. As such its study in-volves an interesting mix of pure and ap-plied mathematics. At the pure end it isthe study of the properties certain spacesof functions. Examples being polynomials,splines radial basis functions and Beziersurfaces. At the applied end it is the con-struction of algorithms to enable e�icientuse of these spaces of functions in practi-cal problems.

Recent applications of Approximation The-ory here at Canterbury include fitting sur-faces to noisy point clouds, applied to

the custommanufacture of artificial limbs,and fitting geophysical data sets such asgold grade measurements from drill holesin mines.

The first part of this course will concen-trate on the fundamentals of approxima-tion of functions of one variable. Centraltopics will be approximation by algebraicand trigonometricpolynomials, and theex-istence, characterisation and uniquenessof best approximations from finite dimen-sional normed linear spaces.

In the latter part of the course we will de-velop somemore recent topic. The exacttopic will be chosen by the class. Exam-ples of possible topics are radial basisfunctions, the use of Bezier surfaces inmodeling, and penalized least squaresand L1 methods for modeling noisy data.

Enquires: Rick Beatson

MATH411 15 pointsTopics in AlgebraMATH411-16S1 (C)This course will provide an introduction toAlgebraic Geometry. This is the study ofsolutions to systems of polynomial equa-tions, for example a curve in the plane de-finedbyanequationx3+y3 = 1. Themeth-ods are mostly algebraic, but the intuitioncomes from geometry. You will developyour ability to translate your geometric in-tuition (which tells you what you shouldbeexpecting) intoprecise algebraic expres-sions (which allow you to cash in on thatintuition). Topics covered will include al-gebraic sets in a�ine and projective space,algebraic curves, elliptic curves, curve sin-gularities, and others as time permits.

Familiaritywith linear andabstract algebra(MATH203 and MATH321) is encouraged.

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Enquires: Brendan Creutz

MATH412 15 pointsUnconstrained OptimizationMATH412-16S2 (C)This course looks at the minimization ofsmooth functions of several variables. Thefirst part of the course examines gradi-ent based methods using line searches,including Newton, quasi-Newton, and con-jugate gradient methods. A selection ofother topics is then introduced, includ-ing trust region methods andmethods forconstrained optimization. Demonstrationso�ware is used to illustrate aspects ofvarious algorithms in practice.

Enquires: Rachael Tappenden

MATH420 15 pointsHilbert SpacesMATH420-16S2 (C)The theory of Hilbert spaces is fundamen-tal in many areas of modernmathematicalanalysis, having a clear and easy-to-graspgeometric structure, just like Euclideanspaces. However, unlike Euclidean spaces,Hilbert spacesmaybe infinite dimensional.The coursewill be self-contained, introduc-ing important spaces (especially L2(m)),operators on them, and basic spectral the-ory. Prior exposure to MATH343 would bean asset, but is not mandatory.

Enquires: Maarten McKubre-Jordens orHannes Diener

MATH427 15 pointsLie Groups and Lie AlgebrasMATH427-16S2 (C)Lie groups are an essential tool in manyareas of mathematics and physics. Theyare o�en found as groups of symmetries

of ‘nice’ mathematical objects like geome-tries or dynamical systems. The most im-portant Lie groups are finite-dimensionaland occur as groups of matrices over realor complex numbers. For example, thegroup SO(3) of all rotations of Euclidean3-space or its closely related groups SU(2)and Spin(3) are Lie groups. One is inter-ested in their properties and how thesegroups can be realised in higher dimen-sions.

Every Lie group has an associated Lie al-gebra which is a very good linear approx-imation of the group. Many properties ofthe Lie group can be deduced from its Liealgebra.

This course gives an introduction to the ba-sic theory of finite-dimensional Lie groupsand their associated Lie algebras and lin-ear representations.

Enquires: Gunter Steinke

MATH429 15 pointsCombinatoricsMATH429-16S2 (C)Matroids (combinatorial geometries) areprecisely the structures that underlie thesolution of many combinatorial optimisa-tion problems. These problems includescheduling and timetabling, and findingthe minimum cost of a communicationsnetwork between cities. Matroid theoryalso unifies the notions of linear indepen-dence in linear algebra and forests in graphtheory as well as the notions of duality forgraphs and codes. This course is an intro-duction to matroid theory and is designedfor mathematics and computer sciencestudents.

Enquires: Charles Semple

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MATH431 15 pointsGraph TheoryMATH431-16S2 (C)In a nutshell, graphs are mathematicalstructures which model relationships be-tween objects. Graph theory is the branchof combinatorics concerned with theirstudy, and has grown to become a veryrich and diverse discipline in its own right.It has applications in almost every scien-tific field, from analysing the spread of epi-demics to modelling social networks.

In this self-contained course we will ex-plore a range of topics from graph theory,considering both theory and applications.The course is intended for students major-ing in Mathematics or Computer Science.Does not require MATH120 or MATH220.

Enquires: Jeanette McLeod

MATH432 15 pointsSpecial Topic in Mathematics(Foundations of Mathematics)MATH432-16S2 (C)An introduction to the philosophy of math-ematics, classical and intuitionistic logic,set theory and Gödel’s theorems. SeeMATH 336.

Enquires: Maarten McKubre-Jordens

MATH433 15 pointsMathematics in PerspectiveMATH433-16S1 (C)What is Mathematics? What are some ofthe keymoments in the history of Mathe-matics? What kinds ofmathematical resultare considered important, and why?

This course is about the history, philoso-phy, people and major results of Mathe-matics over the centuries. Since we will

minimise the attention paid to technicaldetails, the course should be accessiblenot only to those with a 200 level Math-ematics background, but also to intellec-tually mature students in Philosophy andrelated subjects. In particular, it is stronglyrecommended for anyone who intendsteaching Mathematics at any level fromprimary school onwards. See MATH380.

Enquires: John Hannah

MATH438 15 pointsSpecial Topic in Mathematics(Advanced Complex VariablesMATH438-16S2 (C)This is a second course in complex vari-ables that introduces the student to somebeautiful results and important applica-tions of complex analysis. Topics cov-ered include: Liouville’s theorem, openmapping theorem, Cassorati-Weierstrasstheorem, argument principle, Rouche’stheorem, maximum modulus principle,Schwarz’s lemma, normal families, Rie-mannmapping theorem. Additional mate-rial (time permitting) may be chosen fromthe following: Infinite products, Runge’stheorem, univalent functions, subhar-monic functions, value distribution theory,Hardy spaces.

Enquires: Ngin-Tee KohMATH439 15 pointsTopics in AlgebraMATH439-16S1 (C)This course in modern algebra. SeeMATH321.

Enquires: Gunter Steinke

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MATH443 15 pointsMetric, Normed and Hilbert Spaces

MATH443-16S1 (C)This course introduces those parts of mod-ern analysis that are essential for manyaspects of Pure and Applied Mathematics,Physics, Economics, Finance, and so on.For example, if you want to understandthe convergence of numerical algorithms,approximation theory, quantummechan-ics, or the economic theory of compet-itive equilibrium, then you will need toknow something about metric, normedand Hilbert spaces. See MATH343.

Enquires: Hannes Diener

MATH449 30 pointsProjectMATH449-16W (C)A whole year research project in mathe-matics (see Honours projects).

Enquires: Mark Hickman

MATH475 15 pointsIndependent course of studyMATH475-16S1 or MATH475-16S2 (C)This course allows a student to perform di-rected reading of a particular topic undera Mathematics lecturer. The topic choice isby mutual arrangement with the lecturer.Students should ensure these arrange-ments are in place before enrolling in thecourse.

Enquires: Any Mathematics Lecturer

MATH491 15 pointsSummer Research ProjectMATH491-16SU2 (C)This 150-hour course provides students

with an opportunity to develop mathe-matical research skills and to extend andstrengthen their understanding of an areaof mathematics.

Enquires: Jeanette McLeod

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STATISTICS

STAT446 15 pointsGeneralised Linear ModelsSTAT446-16S1 (C)How do you analyse data that does notfit the standard methods such as ANOVAand regression? How do you deal withdata that are very non-normal, are countsrather than measurements, are corre-lated and have interdependencies? In thiscourse we introduce you to the very use-ful toolbox of Generalised Linear Models(GLMs). This is a natural progression fromunderstandingANOVA, regressionandmul-tivariate techniques. We will learn aboutthe general framework for GLMs, and howto use GLMs for analysing data. We willintroduce you to the package R, and willuse this so�ware throughout the course.Some background in statistical analysismethods is necessary, and useful coursesto have completed are STAT220, STAT212and STAT315. No experience in R is neces-sary. See STAT319.

Enquires: Jennifer Brown

STAT447 15 points(O�icial Statistics)STAT447-16S2 (C)This course provides an overview of thekey areas of O�icial Statistics. Topics cov-ered include data sources (sample sur-veys and administrative data); the legaland ethical framework of o�icial statistics;an introduction demography; the collec-tion and analysis of health, social and eco-nomic data; data visualisation includingpresentation of spatial data; data match-ing and integration; the system of NationalAccounts.

Enquires: Jennifer Brown

STAT449 30 pointsProjectSTAT449-16W (C)A whole year research project in Statistics(see Honours projects).

Enquires: School of Mathematics andStatistics Reception

STAT450 15 pointsAdvanced Statistical ModellingSTAT450-16S1 (C)This course provides an introduction to arange of statistical techniques used in theanalysis of spatial data. It will cover thebasic concepts and techniques of spatialdata analysis (SDA) and provide a widerange of applications examples from vari-ous fields such as geology, demographics,epidemiology and environmental sciences.A comprehensive lab programme uses avariety of so�ware packages (includingArcGIS, Geoda, geoR andWinBUGS) to ex-plore and analyse spatial data using thetechniques taught in the course.

Enquires: Elena Moltchanova

STAT455 15 pointsSampling MethodsSTAT455-16S1 (C)This course looks at practical methods forgathering new data, the raw material ofStatistics. See STAT312.

Enquires: Blair Robertson

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STAT456 15 pointsTime Series and Stochastic ProcessesSTAT456-16S2 (C)Hereweexplain some techniques tomodelobservations taken sequentially over time.This kind of data is very common in Biol-ogy, Environmental Sciences, Economicsand Finance. Time series methods arewidely used for forecasting. This course isapplication oriented, and computers areused to analyse real time series data. SeeSTAT317.

Enquires: Marco Reale

STAT460 15 pointsExtreme Value StatisticsSTAT460-16S1 (C)This course aims to develop the theoryandmethods for modelling the extremesof random processes. Extreme value the-ory moves away frommore traditional sta-tistical techniques where the aims are tomodel the usual (or in some sense aver-age) behaviour, to consider the unusualor rare events. It has received wide ap-plication in many fields where the risk as-sociated with rare events are of concern,e.g. finance/economics, hydrological mod-elling, climate change, engineering (struc-tural design) andmaterial science (mate-rial fatigue/failure).

The course will cover the mathematics un-derlying extreme value models, statisticalinference using likelihood and applica-tions to real data, with implementation inthe so�ware package R. Recommendedpreparation includes secondyearStatistics(preferably STAT214) and at least full firstyear Mathematics (MATH103 or EMTH119or equivalent).

Enquires: Carl Scarrott

STAT461 15 pointsBayesian InferenceSTAT461-16S2 (C)This course explores the parametricBayesian approach to Statistics by con-sidering the theory, methods for comput-ing Bayesian solutions, and examples ofapplications. The key advantage of theBayesian approach is that it naturallyprovides probabilistic measures of uncer-tainty along with the inference. Topicsthat may be covered include: Theoreticalfoundations of Bayesian Statistics, choiceof prior distributions, Bayesian estima-tion and credible regions, Bayesian testsand model selection, methods for com-puting Bayesian solutions, hierarchicalmodels, and applications to linear models.Students should have a soundmathemati-cal background and a good foundation inStatistics and Probability, at least up to thelevel of STAT213 or STAT214. See STAT314.

Enquires: Elena Moltchanova

STAT462 15 pointsData MiningSTAT462-16S2 (C)Data mining refers to a collection of toolsto discover patterns and relationships indata, especially for large data bases. It in-volves several fields including data basemanagement, statistics, artificial intelli-gence, and machine learning, and it hashad a considerable impact in business, in-dustry and science.

This course provides an introduction tothe principal methods in data mining:data preparation and warehousing, super-vised learning (tree classifiers, neural net-works), unsupervised learning (clusteringmethods), association rules, and the deal-ing with high-dimensional data (PCA, ICA,

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multidimensional scaling). Students willsee applications from various fields, suchas commerce (fraud detection, productplacement, targeted marketing, assessingcredit risk) andmedicine (diagnostics). Wewill use data mining so�ware to illustratemethods with data sets from these fields.

Students must (i) do problems that areassigned throughout the term and (ii) re-search an area and write an account ofit; the instructor will give suggestions fortopics in class. See STAT318.

Enquires: Blair Robertson

STAT463 15 pointsMultivariate Statistical MethodsSTAT463-16S2 (C)Multivariate statistical methods extractinformation from datasets which consistof variables measured on a number of ex-perimental units. The application of thesemethods is blooming with the availabilityof large datasets from a wide range of sci-entific fields, combined with the adventof computing power to implement them.Examples abound in fields as diverse asbioinformatics, internet tra�ic analysis,clinical trials, finance andmarketing. Thiscourse will cover the theory and appli-cations of various multivariate statisticalmethods. See STAT315.

Enquires: Daniel Gerhard

STAT470 15 pointsSpecial Topics in Statistics(Advanced Time Series Methods)STAT470-16S2 (C)In many applications, in particular in fi-nance and economics, observed data se-rieso�enexhibit abehaviourwhichcannotbemodelled with linear time series mod-

els (i.e. ARMA processes). Thus alternativemodels allowing for a nonlinear behaviourare called for and are successfully used.For instance, Robert Engle was awardedthe Nobel Prize in 2003 for introducing theso-called (G)ARCHmodel. In this coursewewill first review some materials on lineartime series methods, then consider andanalyse several classes of nonlinear timeseries models, such as GARCH, Markov-switching as well as threshold autoregres-sive time series models. We study theircommon probabilistic and statistical con-cepts and theory (Markov chains with un-countable state space, stochastic recur-rence equations, ergodicity and mixing).Finally, we will derive and apply estima-tors for the model parameters.

Enquires: Marco Reale

STAT471 15 pointsSpecial Topic in Statistics(Big Data)STAT471-16S1 (C)Big Data refers to the large and o�en com-plex datasets generated in the modernworld: data sources such as commercialcustomer records, internet transactions,environmental monitoring. This courseprovides an introduction to the theory andpractice of working with Big Data and willbe of interest to any student wanting toprepare themselves for work industry, orfurther research.

The course will give you an understandingof the basic concepts of how and whereBig Data arises. By taking the course youwill be able to describe data structuresandmechanisms for the capture, storage,processing, summary and visualisation ofBig Data; implement practical methods fordata acquisition and management using

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appropriate so�ware (SQL, R, Python, Perl;Hadoop, Mapreduce); and, understandbasic methods of analysis of Big Data, in-cluding methods from machine learningfor high dimensional data.

Enquires: Jennifer Brown

STAT472 15 pointsSpecial Topic in Statistics(Advanced Data Analysis and StatisticalConsulting)STAT472-16S2 (C)In most undergraduate courses, you aretaught the theory behind a method andthen given neat examples to which it canbe applied and so�ware to apply it. In re-ality, the most common question you willhear from a non-statistician is ’how do Ianalyse my data? So you are the one whohas to come up with the appropriate re-search question and choose the suitablemethod (and sometimes learn it quicklytoo) . It is common in real world applica-tions for the experiments have not beenwell planned and for data to be missing,which will need to be taken it into account.The assumptions underlying the statisti-cal model (e.g. homoscedasticity and nor-mally distributed) o�en do not hold andyou will have to knowwhat to do. Finally,your fellow scientists, laymen and policy-makers are all interested in di�erent as-pects of the research question and that israrely the statistical significance of yourANOVA: you have to know how to commu-nicate your results clearly, correctly ande�iciently and how to defend your choicesin data analysis and collection.

This course is about the reality of being anapplied statistician. Besides covering theabove points in class, individual statisticalconsulting session will provide you with

hands-on experience.

Good knowledge of multivariate statisticalmethods, GLMs, and basic sampling the-ory expected. Working knowledge of R isrecommended.or forecasting methods. Itprovides extensive training in forcastingandmodelling techniques such as smooth-ing, dynamic regressions, multivariate au-toregressions, state space models, andneural networks with a wide range of ap-plications.

Enquires: Elena Moltchanova

STAT474 15 pointsSpecial Topic in Statistics(Scalable Data Science)STAT474-16S1 (C)Scalabledata science is all about analysing’big data’, real-world datasets that do notfit into any single computer. So one needsa cloud-computing platform, a manageddistributed computing environment madeupof anetworkof several tensor hundredsof computers, to analyse such big data.This advanced undergraduate course in-volves a flipped class-roomof the archivededX course on scalable machine learning.These edX lectures and the accompany-ing labs using Apache Spark will be sup-plemented by weekly tutorials and discus-sions of research papers underpinning thecore ideas of Spark. This independentcourse of study can form the groundworkfor an honours project in scalable data sci-ence that is customized to the interestsand skills of the student.

More details can be found here: http://www.math.canterbury.ac.nz/~r.sainudiin/courses/ScalableDataScience/

Enquires: Raazesh Sainudiin

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STAT475 15 pointsIndependent course of studySTAT475-16S1 or STAT475-16S2 (C)This course allows a student to perform di-rected reading of a particular topic undera Statistics lecturer. The topic choice is bymutual arrangementwith the lecturer. Stu-dents should ensure these arrangementsare in place before enrolling in the course.

Enquires: Any Statistics Lecturer

STAT491 15 pointsSummer Research ProjectSTAT491-16SU2 (C)This 150-hour course provides studentswith an opportunity to develop statis-tical research skills and to extend andstrengthen their understanding of an areaof statistics.

Enquires: Jeanette McLeod

HONOURS PROJECTS

A range of possible projects are outlined.However, this list is not exhaustive andother possibilities for projects are certainlypossible.

Project supervision is by mutual agree-ment of the supervisor and student. Youshould arrange your project by the end ofthe first week of term in 2016. It is sug-gested that you seek out possible super-visors during the enrolment week (or be-fore if you have a clear idea of what sort ofproject interests you).

You will hand in a written report on theproject by Monday September 15, 2016;part of your project assessment will be anoral presentation during Term 4.

PROJECTS IN MATHEMATICS

Algorithms for tangent plane continu-ous surface fittingRick Beatson

This project considers the fitting of tangentplane continuous surfaces to points andnormals data sets. The end goal is to ef-ficiently model underground rock forma-tions, and other blobby shapes, occuringin applications. The project will build onan existing algorithm which uses quinticBezier elements.

There are diverse aspects that could beconsidered. An algorithmic aspect is toconsider and experiment with some localoptimisation steps, which would be usedto improve a given fit when the normalsare noisy.

Another aspect is to recast the current the-

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ory in terms of a di�erent parametrisationof the Bezier elements. This would quiteprobably give rise to a cleaner theory, andprobably a more e�icient implementation.

Finally, on the implementation end, a coretask in the current code is to solve a sparsesemi-definite system. The current solveris crude. The theory and implementationof a more e�icient solver would be an in-teresting task for someone who enjoysnumerical analysis, and wants to experi-ment with parallel algorithms.

Arithmetic of Elliptic CurvesBrendan Creutz

An elliptic curve is a geometric object witha rich algebraic structure. There is a natu-ral way to ‘add’ two points on the curve toget a third, resulting in an algebraic group.One of themost prominent openproblemsinmathematics seeks to obtain a better un-derstandingof the structureof suchgroups(cf. The Birch and Swinnerton-Dyer Con-jecture at http://www.claymath.org/millennium-problems).

These groupshavealsobegun toplay an in-creasing prevalent role in cryptography, asthey provide cryptosystems that are moresecure and e�icient than traditional sys-tems based on integer factorization or dis-crete logarithms in finite fields.

This project will introduce the student tothe theory of elliptic curves by first inves-tigating the classical geometry of ellipticcurves over the real or complex numbers.Depending on student interest, the projectwill then explore the situation over therational numbers, function fields, or finitefields, as well as investigating potentialapplications.

A Mathematical Approach to OptimalFilteringMiguel Moyers Gonzalez, Rua Murray andPhil Wilson

Porous media are materials composedof a network of pores in a solid matrix.They are important for a large range ofreal life applications in the oil, gas andprocess industries, structured lightweightmaterials, and studies of flow properties.3D printers o�er the opportunity to con-trol the size, shape and location of thevoids in the porous morphology. The goalof the project is to develop and study amathematical approach to the di�erentgeometrical morphologies that can be the-oretically designed in porous media. Theproblem can be formulated as an optimalcontrol problem and will be studied usingMATLAB or an equivalent.

RandomDynamical SystemsRua Murray

Random dynamical systems are obtainedby a coupling a known family of determin-istic systems to a driving stochastic pro-cess. The resulting behaviour is one kindof “non-autonomous dynamics”. For de-terministic dynamics, questions as simpleas “how do you define an attractor?” havefairly straight-forward and unambiguousanswers. However, in the random setting,the answers are more complicated andinteresting. This project will investigateergodic properties of random dynamicalsystems, focussing particularly on whatinsights “random transfer operators” cangive about the behaviour of random dy-namical systems. Some numerical calcula-tion in Matlab (or similar) will be required.Applications include mass transfer/mixingin nonlinear fluid flows.

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Synchronisation and Coupled Dynami-cal SystemsRua Murray

This project will begin with a review of thewell-studied and well understood topicof synchronisation in dynamical systems.The basic setup is that two or more identi-cal systems are allowed to run from di�er-ent initial conditions, but a small amountof coupling is present between them. Evenwhen the dynamics is chaotic, orbits ofthe various oscillators can converge toa synchronised state. A�er learning themathematics behind this phenomena, theproject could proceed in any of severaldirections: studying chaotic synchronisa-tion to encrypt communications; investi-gating the e�ects of multiple-timescaleson synchronisation; exploring “approxi-mate synchronisation” between similar,but non-identical systems. Applicationsinclude oscillatory behaviour and signalpropagation in (excitable) biological tis-sues.

Linear ProgrammingChris Price

This project looks at applications of dis-crete linear programming and solutionmethods via linear programming andother relaxations. Math303 is requiredas a prerequisite.

Computational Optimization and Appli-cationsRachael Tappenden

Optimization plays a crucial role in manymodern, real-world applications, fromfinance to medical imaging, and fromscheduling problems to big-data. Belowis a list of several projects that studentsmay be interested in working on.

• Portfolio Optimization: Given someinitial budget, and a basket of finan-cial instruments (stocks, bonds, etc),how should one allocate their fundsin order to maximize their expectedprofit? Two questions that onemaywish to answer are: (i) What kinds ofalgorithms can be used to e�icientlysolve such problems, and (ii) Therearemanyways ofmodeling portfoliooptimization problems, but what is‘best’?

• Medical Imaging and Optimization:For many imaging modalities, in-cluding Computed Tomography (CT)and Magnetic Resonance Imaging(MRI), one must solve an ‘inverseproblem’ to obtain an image. In thisproject we will investigate CT imagereconstruction. We will see that theCT ‘inverse problem’ o�en involvesthe solution of a large linear systemof equations, and the student will in-vestigate algorithms for solving suchsystems.

• There are many other projects thatmay be of interest to students. Gen-eral topics include: solving largescale linear systems e�iciently, algo-rithms for determining the eigenval-ues of large, sparse matrices, and al-gorithms for big-data problems. Seeme for further details.

For these projects, it would be helpful ifthe student had some knowledge of Mat-lab and of applied linear/matrix algebra.

Computing the InfiniteHannes Diener

Since most mathematical objects areideal/infinite, but computers can only ever

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deal with discrete/finite objects, it seemsthat we almost always have to confineourselves to using approximations andshadow types (such as ‘float’ instead ofthe reals, or ‘int’ instead of the naturalnumbers) when transferringmathematicalideas to a computer.

Surprisingly though, it is possible to com-pute with (a large class of) infinite subsetsof natural numbers, real numbers, and soon, not just as approximations, but as theactual objects. In this project a studentshould investigate some of the mathemat-ical background of such an approach, andideally implement some of the ideas. De-pending on the student’s background theproject can either focus on the theoreti-cal mathematical foundations, or on thepractical implementation. For the firsta background in logic is welcomed (forexample MATH130, 230, or 336), for thesecond some familiarity with a program-ming language which allows elements of‘functional programming’ (such as Python,Haskell, C++) is useful.

Philosophy and Foundations of Mathe-maticsMaarten McKubre-Jordens

What is mathematics? What is a mathe-matical theory? What are the objects ofmathematical thought? Why is mathemat-ics, an abstract subject, so e�ective in thereal world? And when a mathematicianproves something, what are they actu-ally doing? In this project you will studydevelopments in the philosophy and foun-dations of mathematics. Depending onyour interest, the project may investigatephilosophical positions concerning math-ematics, mathematical truth and provabil-ity, or the limits of mathematics.

Non-classical MathematicsMaarten McKubre-Jordens

Why are some mathematical questionsnot just unanswered, but unanswerable?And what is so bad about contradictorytheories? The last few decades have seena burgeoning in non-classical logics. Inthis project, you will investigate the re-lationship of these logics to mathemat-ics. You will discover that there are manymore facets to mathematics than many -even practicing mathematicians - realise.Moreover, you will learn some newmath-ematics non-classically, and gain insightinto distinctions that cut mathematics atits very joints: the limits of provability, orthe meaning of mathematical statements.Depending on your interest, the projectmay take you into: the emergence of log-ics frommathematics; the application oflogics to mathematics; constructive math-ematics; or paraconsistent mathematics.

History of MathematicsJohn Hannah

The second half of the 19th century andthe first half of the 20th century saw theintroduction and development of severalquite abstract branches of mathematics.Examples includeabstract algebra (groups,fields, rings, ideals, modules), functionalanalysis, topology, and category theory.In this project you will follow this develop-ment in a topic of your own choosing, andin the process you will both learn somenewmathematics, and try to discern theinfluences (mathematical, philosophicalor cultural in origin) which led to thesedevelopments.

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Ecological NetworksAlex James

FromtheAmazon rainforest to theOkeoverstream species interact with each otherin complex systems. These interactionscome in many forms: competitive - theyboth compete for the same resource,predator-prey - eat or be eaten, mutual-istic - both benefit from the relationship.Howdoes the architecture of these interac-tions a�ect the behaviour of the system?Should a species interactwith asmanyoth-ers as possible? Should they form groupsor cliques? Is there a conflict betweengood for an individual and good for thesystem?

Useful pre-requisites for this projectcould include dynamical systems, discretemaths and some Matlab or other com-puter programming. An interest in ecologyis also helpful.

I also have other projects available thatcombine maths and ecology. Please con-tact me to find out more.

Goannamicrohabitat usageAlex James & Daniel Gerhard

Australian Goanna are subject to a rangeof predators (mostly introduced!). In thisproject you will use new data collected byresearchers at LandcareResearch tounder-stand how Goanna respond to predatorsin di�erent habitats.

This project will give you an opportunity todo novel research on real ecological data.The project will have a co-supervisor fromLandcare Research giving you a chance tomake important contacts there and learnhow data analysis is done outside a univer-sity environment.

Modelling Collective Cell MovementMike Plank

Collective cell behaviour is the drivingforce behind many physiological pro-cesses, including embryonic development,tissue repair and tumour growth. Exper-iments on collective cell behaviour typi-cally collect data at the level of the pop-ulation rather than the individual cell.WeâĂŹd like to be able to translate datafrom observing populations of cells intoknowledge about how individual cellswork and how they interact with theirneighbours. This project will approachthis problem using approximate Bayesiancomputation (ABC). At its simplest, this in-volves sampling model parameters froma prior distribution and simulating cell be-haviour. If themodel output is âĂIJcloseâĂİto the experimental data, the parametervalues are accepted as part of the posteriordistribution, otherwise they are rejected.This will be used to estimate quantitiessuch as cell proliferation and movementrates and the strength of interactions withneighbouring cells.

This project will require some experienceof a computer programming language, e.g.Matlab and an interest in working with bio-logical data. Prior knowledge of Bayesianstatistics is NOT required.

Smaller Fish to Fry?Mike Plank

Modern fisheries management is almostuniversally based on the principle of pro-tecting small fish from capture and target-ing large fish. Fisheries that do no con-form to this dogma, such as small-scaleAfrican fisheries, are seen as destructive.But mathematical modelling and empir-ical experience has shown that, actually,these fisheries can be more sustainable

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than you might expect. The aim of thisproject is to investigate the optimal sizeof fish to catch from the point of view of anindividual fisher. This is a game theoreti-cal problem, because the choices madeby one fisher can a�ect the fish popula-tion, and therefore the outcomes for otherfishers. Of primary interest is the ques-tion: if each individual fisher behaves soas to maximise his/her own yield, whatdoes the aggregate fishing pattern looklike across the whole fishery? Are regula-tions needed to stop the fishers driving thefish population to extinction and/or driv-ing each other to economic penury? Orcan a self-organising group of individualfishers be ecologically and economicallysustainable?

This project will require some experienceof a computer programming language, e.g.Matlab, and an interest in using mathe-matical models to investigate problems inecology, sociology or economics.

Tree-Child NetworksCharles Semple

Phylogenetic networks generalise phylo-genetic (evolutionary) trees by allowingfor non-tree-like evolutionary events suchas lateral gene transfer and recombina-tion. Themathematical study of phyloge-netic networks is recent and, arguably, nomore more that fi�een years old. Amongstthe many subclasses of phylogenetic net-works being studied, tree-child networksare emerging as an increasingly importantclass. What makes them so important?Howdo they relate to other classes? In thisproject, we investigate these questions.There are no prerequisites for the project.

Negative CorrelationCharles Semple

It follows from the work of Kirchho� (1847)that the spanning trees of a connectedgraphG are negatively correlated. That is,for edges e and f ofG, it is more likely thata spanning tree chosenat randomcontainse than one that contains e knowing thatit also contains f . This seems intuitivelyclear but, nevertheless, still requires proof.What if, instead of choosing a spanningtree, we choose a forest at random? Doesthe analogous result still hold? The pur-pose of this project is to explore this andrelated problems. While some knowledgeof graph theory would be helpful, it is nota prerequisite for the project.

The Banach-Tarski paradox and otherparadoxical constructionsGunter Steinke

Can you have your cake and eat it too? Itseems quite simple: cut your cake in lit-tle pieces and reassemble them into twocakes each the size of the original one.Then you can eat one and keep the other.If this sounds too good to be true it is–forreal life cakes that is. However, for mathe-matical cakeswith all the intellectual nour-ishment they provide this can be done.

The project explores what is known asthe Banach-Tarski Paradox: constructionsin which sets in Euclidean space can bepartitioned into subsets which, a�er cer-tain transformations, can be rearrangedto yield sets with very di�erent properties.For example, the project looks at sets thatwhen reassembled produce two copies ofthemselves and ways to reassemble a diskinto a square of equal area.

Onemay also look at other constructionswhich are contrary to intuition. For ex-

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ample, one usually thinks of curves as1-dimensional objects and somethingwhose graphs can be drawn; they shouldbe di�erentiable except for a few points.However, one can actually fill the entireunit square (a 2-dimensional object) as thecontinuous image of the unit interval. Theproject explores how such space-fillingcurves can be constructed and what prop-erties they must have.

Regular tilings and crystallographicgroupsGunter Steinke

A tiling (or tessellation) of the plane is acovering by non-overlapping polygons liketiles in a parquet floor extended infinitelyin all directions. The most symmetric andaesthetic tilings are the regular ones inwhich all polygons involved are congru-ent and regular. Similar configurationscan be constructed on the sphere, whichthencorrespond to thePlatonic solids, andother surfaces. The symmetries of thesetilings lead to groups which can be usedto describe and characterise the tilingsfromwhich the groups arise. In case of Eu-clidean 3-space one obtains the so-calledcrystallographic groups, which describethe symmetry patterns of crystals, but theterm crystallographic group is also used incase of patterns in the plane.

This project explores the concepts of sym-metry groups and crystallographic groupsinvestigates how they are used to obtain(and classify) all regular tilings of the planeand how to create some tilings on othersurfaces.

The Mathematics of Natural DisastersPhil Wilson & Miguel Moyers-Gonzales

Geologic processes can adversely impact

human life and infrastructure, represent-ing a significant risk to national secu-rity. For example, volcanic ash can chokeemergency generators, pollute reservoirs,and damage hydroelectric turbines. Thisproject will model selected aspects of nat-ural disasters using mathematical and nu-merical techniques. The aim is to betterunderstand the underlying physical pro-cesses in order to mitigate their conse-quent adverse e�ects. The project is partof a broader inter-disciplinary researchgroup. The ideal candidate would have agood background in PDEs andMATLAB, butall applicants will be considered.

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PROJECTS IN STATISTICS

Environmental StatisticsJennifer Brown

Environmentalmonitoring is a fastmoving,and important field of research. Data onenvironmental processes such as changesin water quality, endangered species dis-tribution, weed invasion, and biodiversityare used to inform and guide howweman-age our environment.

One use of environmental data is to buildmodels to predict species distribution,and to predict the e�ect of environmen-tal changes. In this project we will lookat di�erent methods used to collect fielddata and the e�ect of these di�erences onpredictionmodels. We will use computersimulations to model data collection andanalysis.

Bayesian Data AnalysisElena Moltchanova

Bayesian statistics o�ers a di�erent worldview and o�en allows for richer andmoreflexible inference than the standard classi-cal techniques.

1. One area where Bayesian approach isparticularly useful is in jump processeswhere an unknown number of abruptchanges might have occured during themonitoring period. Reversible JumpMarkov Chain Monte Carlo (RjMCMC) isone of the methods applicable in suchsituations. The project will concentrateon construction and implementation ofthis algorithm to a long-termmultivariatetime series. Good R programming skillsand previous familiarity with Bayesian In-ference (STAT314/STAT461) required. (Noprior knowledge of time series analysis isnecessary)

2. Estimation of parameters of stochas-tically truncated distributions is anotherarea where Bayesian approach is useful.Instead of having a clear cut-o� points,samples from stochastically truncated dis-tributions include observations from theoriginal non-truncated distribution withsome probability. The estimation is thusconcerned not only with the original distri-bution but alsowith the inclusion function.Reconstruction of historical distribution ofpopulation heights from the records avail-able for army and navy recruits is one areaof interest to economists, social scientistsand anthropologists and forms the focusof this project. Good R programming skillsand previous familiarity with Bayesian In-ference (STAT314/STAT461) required.

Extreme Value Statistical ModellingCarl Scarrott

The statistics youmeet on undergraduatecourse typically focus on the capturing the"usual" characteristics of a process (e.g.properties of mean, median or variance).Extreme value statistics focus on under-standing the unusual or rare events of aprocess. Extremesareof interests inall sortof di�erent fields, for example:

1. financial risk (e.g. estimating Valueat Risk for risk and portfoliomanage-ment)

2. engineering (e.g. designing struc-tures to withstand the stongestforces they could be exposed to, e.g.wind exposure on bridges or forceson buildings due to earthquakes!)

3. environment (e.g. are Christchurch’swinter air pollution extremes reduc-ing due to government interven-tions?)

4. IT (e.g. can our servers and network

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cope with peak demand when yourassignments and projects are duein?)

I have various honours projects in extremevalue modelling for financial, environmen-tal, medical and industrial applications.So if you are thinking of becoming a statis-tical "extremist" then get in touch.

Beloware twoprojectswhichareaccompa-nied with a $1000 stipend. Students withwork with Dr Hamish Jamieson from theOtago Medical School/Canterbury DistrictHealth Board and an academic sta� mem-ber from our School.

Canterbury District Health BoardDataset

PROJECT ONETo assess the use BlueFern supercomputerat UC to analyse the a large dataset fromtheCanterburyDistrictHealthBoardonpa-tient health. There are many factors thatcan be explored andmodelled, and all areuseful for the CDHB, e.g., rest home admis-sion, mortality, hospital admissions. Thedataset is part of a nationwide, and inter-national, programme to assess health inthe ageing population.

PROJECT TWO (With the Brain Research In-stitute)To use the Canterbury District HealthBoard dataset on health in the ageing pop-ulation to identify the characteristics ofpeople with Parkinson’s Disease and pos-sible predictors of the disease. The secondpart of this project will look at what con-founding factors influence the outcomesin patients with cognitive impairment.

For more details, please contact JenniferBrown.

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University of CanterburyTe Whare Wananga o WaitahaPrivate Bag 4800Christchurch 8140New Zealand

Telephone: +64 366 7001Freephone in NZ: 0800 VARSITY (0800 827 748)Facsimile: +64 3 364 2999Email: info@canterbury.ac.nzwww.canterbury.ac.nz